sm 05 phase relationship

Soil Mechanics ICE‐222CE 222

Chapter 2: Phase relationship of soils

1

Three phases of soil 2

Soil element  Three phases

Air VW = 0

in natural state of the soil element

ume

= V

. = W

Air

Water Vw

VaVv

VWw

Wa = 0To

tal v

olu

Tota

l wt.

Solids V

Vw

V

V

W

wW

To Solids VsVsWs

Weight‐Volume relationships 3

Weight Volume

Air Va

V

Wa = 0

Water Vw

Vv

VWw

W VW

Solids VsVsWs

Weight relationships 4

The common terms used for weight relationships 

are moisture content, wmoisture content, w

and unit weight γunit weight, γ. 

Moisture content, w 5

Weight

Moisture content is also known as water content, w, is expressed either in terms of weight or mass.

W

AirWa = 0

Weight%100×=

s

w

WWw

WaterWwW

%100×=s

w

MMw

SolidsWs

A small w indicates a dry soil, while a large w indicates a wet one.

Usual values: 3 to 70%

Moisture content values greater than 100% are found in soft soils below ground water table.

Unit weight, γ 6

Geotechnical engineers often need to know the unit weight, γ:

W=γ

V=γ

Two variations of unit weight are commonly used, the dry unit weight, γd, and  the unit weight of water, γw:

Ws wW

Vs

d =γw

ww V=γ

Normally we use γ = 9 81 kN/m3 = 62 4 lb/ft3 for fresh water andNormally we use γw = 9.81 kN/m = 62.4 lb/ft for fresh water, and γw = 10.1 kN/m3 = 64.0 lb/ft3 for sea water

Th i i h f il b l h d bl (GWT) iThe unit weight of soil below the ground water table (GWT) is called saturated unit weight, γsat.

7

Volume relationships 8

The common terms used for volume relationships 

are void ratio, e

andand porosity, n

andanddegree of saturation, S

Void ratio, e 9

Void ratio, e, is defined as the ratio of the volume of voids to the volume of solids.

Air Va

Volume

s

vVVe =

Water Vw

Vv

V

s

Densely packed soils have low void ratios

Solids V Vs

Vhave low void ratios. 

Typical values in the field  Solids Vs syprange from 0.1 to 2.5.

Porosity, n 10

Porosity, n, is defined as the ratio of the volume of voids to the total volume.

Air Va

Volume

VVn v=

Water Vw

Vv

V

V

Typical values in the field f

Solids V Vs

Vrange from 0.09 to 0.7.

Solids Vs s

Degree of Saturation, S 11

Degree of saturation, S, is the percentage of the voids filled with water.

V

Air VaWa = 0

Weight Volume%100×=v

w

VVS

h l f

Water Vw

Vv

VWw

W

S has max. value of 100% when all of voids are filled with water. Such 

Solids V Vs

V

Ws

Wsoil are called saturated soils.

Solids Vs sWs

S values above GWT are usually 5 to 100%. 

S = 0 is found in very arid areas.

e – n relationship 12

nVV

VV

VVevv

vv =====n

VV

VVVVVV

evvvs −=

−=

−=

−==

11

eVV

VV

VV s

v

s

v

vve

VV

VVVVVV

n

s

vs

s

vss

vs

vv+

=+

=+

=+

==11

ne = en =n

e−

=1 e

n+

=1

γ – γd relation13

WWW ws ⎥⎤

⎢⎡

+

Moisture content, w

VWW

W

VWW

VW ss

sws

⎥⎦

⎢⎣

+=

+==γ

D it i ht

( ) ( ) ( )wwVW

VwW

dss +=+=

+= 111 γ

Dry unit weight , γd

VV

( )1( )wd += 1γγ

γ( )wd +

=1γγ

Specific gravity of solids, Gs14

The specific gravity of any material is the ratio of its density to that of water. 

In case of soils, we compute it for the solid phase only, and express the results as the specific gravity of solids, Gs:

ssss V

WVWGγγ

==wsw V γγ

This is quite different from the specific gravity of the entire soil hi h ld i l d lid d i h f dmass, which would include solid, water, and air. Therefore, do not 

make the common mistake of computing Gs as γ/γw.

For most of soils, Gs is from 2.60 to 2.80. 

Specific gravity of solids, Gs15

Typical values of e, w and γd16

γ‐e‐w relationship 17

ws

ss V

WGγ

= wsss VGW γ=Vs = 1

wss GW γ=wsγ

s

wWWw = sw wWW = wsw wGW γ=

Weight

Air

Volumes

ww

WVγ

=Air

Water Vw = wGs

Vv = e

Ww = wGsγw

wγ

sws wGwG==

γ

S lid

w s

V = 1+eW

swγ

wGV =Solids Vs = 1Ws = Gsγwsw wGV =

γ‐e‐w relationship 18

VW

=γeWW ws

++

=1

γe

wGG wsws++

=1

γγγ

( )eGw ws+

+=

11 γγ ( ) e

Gw

ws+

=+ 11

γγe

G wsd +=

1γγ

Weight

Air

Volumee+1

G wsγ Air

Water Vw = wGs

Vv = e

Ww = wGsγw

ews

d +=

1γγ

lid

w s

V = 1+eW

1−= wsGe γSolids Vs = 1Ws = Gsγw

1d

eγ

γ‐e‐w relationship 19

wVS wGS s wGe s= wGSe =v

wV

S =e

S s=S

e = swGSe =

Weight

Air

Volume

wGSe = Air

Water Vw = wGs

Vv = e

Ww = wGsγw

swGSe =

lid

w s

V = 1+eW

SwGe s=

Solids Vs = 1Ws = GsγwS

γ‐e‐w relationship 20

swGSe =As we know

d W WW ws + wGG wsws + γγandVW

=γeWW ws

++

=1

γe

wGG wsws++

=1

γγγ

Weight

Air

Volume

eeSG wws

++

=1

γγγAir

Water Vw = wGs

Vv = e

Ww = wGsγw

( )eSG ws +=1

γγ

lid

w s

V = 1+eW

e+1γ

Solids Vs = 1Ws = Gsγw

γ‐e‐w relationship (fully saturated) 21

wss GW γ= swGSe = At fully saturated conditionS = 1

wsw wGW γ= ww eSW γ= ww eW γ=

WW +Weight

V = V = e

VolumeV

WW swsat

+=γ

WaterVv = Vw = e

V = 1+e

Ww = eγw

eeG wws

sat ++

=1

γγγ

Solids V = 1

V = 1+e

W( )eG wssat

+=

γγ Solids Vs 1Ws = Gsγwesat +1

γ

γ‐n‐w relationship 22

sW

If V = 1, n = Vv/V   => Vv = n V = Vv + Vs =>   1 = n + Vs =>   Vs = 1–n 

( )ws

ss V

WGγ

= wsss VGW γ= ( )nGW wss −= 1γ

Weight

Air

Volumesw wWW =

Air

Water

Vv = n

Ww = wGsγw(1–n))1( nwGW wsw −= γ

S lidV = 1

Solids Vs = 1-nWs = Gsγw(1–n )

γ‐n‐w relationship 23

( ) ( )1

11 nwGnGV

WWVW wswsws −+−

=+

==γγγ

( )( )wnG ws +−= 11γγ

Weight

Air

Volume

Air

Water Vw = wGs

Vv = n

Ww = wGsγw(1–n)

Solids V 1V = 1

W G (1 ) Solids Vs = 1-nWs = Gsγw(1–n )

γ‐n‐w relationship 24

( ) ( )nGnGVW

wswss

d −=−

== 111 γγγ

( )nG wsd −= 1γγ

Weight

Air

Volume

Air

Water

Vv = n

Ww = wGsγw(1–n)

Solids V 1V = 1

W G (1 ) Solids Vs = 1-nWs = Gsγw(1–n )

γ‐n‐w relationship (fully saturated) 25

As we know that, S = 1 at saturation swGe = swG

nn

=−1

( )nGW wss −= 1γ ( ) sGnwn −= 1

Weight Volume ( ) sww GnwW −= 1γ

Water

Vw = wGs

Vv = nWw = nγw

nW γ=

S lid

w s

V = 1

ww nW γ=

Solids Vs = 1-nWs = Gsγw(1–n )

γ‐n‐w relationship (fully saturated) 26

( )1

1 wwswssat

nnGV

WWVW γγγ +−

=+

==

( )[ ] wssat nnG γγ +−= 1

Weight Volume

Water

Vw = wGs

Vv = nWw = nγw

S lid

w s

V = 1Solids Vs = 1-nWs = Gsγw(1–n )

γ‐n‐w relationship (fully saturated) 27

( )nGn

WWw ww

sat −==

1γγ

( )sat Ge

nGnw =−

=1( )nGW wss 1γ ( ) ss GnG 1

Weight Volume

Water

Vw = wGs

Vv = nWw = nγw

S lid

w s

V = 1Solids Vs = 1-nWs = Gsγw(1–n )

Summary 28

Relative density 29

Relative density (Dr) is commonly used to indicate the in‐situ denseness or looseness of granular soils.

ee

minmax

max

eeeeDr −

−=

Relative density 30

eG ws

d +=

1γγ 1−=

d

wsGeγγ 1

(min)max −=

d

wsGeγ

γ

1min −= wsGeγ

γ

(max)dγ

S b tit ti d i t ti f DSubstituting e, emin, and emax into equation of Dr

11

minmax

max

eeeeDr −

−= ⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

−

−=

d

d

dd

ddddrD

γγ

γγγγγγ (max)

(min)(max)

(min)(min)

11dd γγ (max)(min)